Delay quantization technique to reduce steering errors in digital beamformers

ABSTRACT

Quantized delay values used in a digital beamformer are selected so as to minimize the derivative of the beamformer pattern. A first set of quantization error values, one for each hydrophone, is determined from the precise delay values and a second set determined by adding a shading factor times the sample period to each member of the first set. Each set of combinations of members of the first and second sets of error values is then examined to determine which set minimizes the derivative of the beamformer pattern. The quantized delay values are then determined by selecting the quantized delay value corresponding to the error value in the set which minimizes the derivative.

This application is a continuation of application Ser. No. 07/052,083,filed May 21, 1987 now abandoned.

TECHNICAL FIELD

The subject invention relates to the technical field of distant objectdetection by such means as sonar

BACKGROUND OF THE DISCLOSURE

1. Field of the Invention

The subject invention relates to sonar systems and, more particularly,to a technique for quantization of delays in digital beamformers used insuch systems.

2. Description of Related Art

In the prior art, beamformers comprising an arbitrary array ofhydrophones are known. Such arrays permit directional sensitivity. Inorder to sense a signal wavefront at a selected arrival angle to thehydrophone line, the output signal of each hydrophone is appropriatelydelayed so that the collective outputs add coherently, in phase. Theresultant output is characterized by a main lobe about a maximumresponse axis in the desired look direction and several side lobes inother directions.

In a digital beamformer, the outputs of the respective hydrophones aresampled by analog to digital converters and fed to a digital processorfor storage and summing. A shading coefficient may also be applied toeach selected hydrophone output prior to summing, in order to reduce thesidelobes of the system.

In digital beamforming systems, the exact values of delay used to steera beam in a given direction must be quantized to multiples of the systemsample interval. Quantization in effect determines which hydrophoneoutput sample will be associated with a given steering angle.

Most existing digital beamformers quantize the time delays by simplyrounding the full precision values. While this approach assures that thedelays are as close as possible to the true values within the availableaccuracy of the beamformer, it does not relate the quantization to theresulting steering angle In fact, for certain array geometries, roundingcan be shown to be a poor choice from the point of view of preservingsteering accuracy. The same is true of delay quantization by truncatingthe exact values, as is done in some systems.

Existing quantization techniques, such as truncation or rounding thuscontribute to deviations from the desired steering angle In systemswhere high accuracy is required, e.g., the target is at a relativelylong distance, it is desirable to eliminate as many errors as possibleto gain steering accuracy. In other systems, a more accuratequantization technique could produce acceptably accurate steering from acourser quantization, thereby reducing the sample rate and hardwarerequirements, and hence the expense of the system.

SUMMARY OF THE INVENTION

It is therefore an object of the invention to improve sonar systems.

It is another object of the invention to improve beamforming systems.

It is another object of the invention to minimize steering error indigital beamforming systems.

It is another object of the invention to provide a method fordetermining delay quantization in a digital beamformer which moreaccurately steers the sonar beam.

It is another object of the invention to relate quantization of delaysin a digital beam forming system to the steering angle.

It is yet another object of the invention to reduce the required samplerate and attendant hardware requirements and cost of a digitalbeamforming system.

The invention provides a technique for the quantization of delays in adigital beamforming system. The individual delays are quantized so as tominimize the first derivative of the spatial response pattern at thedesired steering angle. Since the location of the zero of thisderivative in the main lobe region defines the location of the truesteering angle, derivative minimization has the effect of minimizingsteering errors due to quantization. The selection of quantized delaysis based on the minimization of a simple weighted sum of quantizationerrors, resulting in very efficient operation.

The invention has potential application in any sonar using a digitaltime delay beamformer. In most systems, the data sample rate issubstantially higher than the Nyquist rate to provide sufficientlyaccurate beam steering. The use of the quantization method of theinvention allows use of a lower sample rate than conventional methods tomeet a given steering accuracy specification. This in turn reducessystem cost through memory and computation rate reductions. In sonarswith a fixed sample rate, the method of the invention produces moreaccurate beam placement than that achieved with conventional delayquantization.

BRIEF DESCRIPTION OF THE DRAWINGS

The just summarized invention will now described in detail inconjunction with the drawings, of which:

FIG. 1 is a schematic diagram illustrating a prior art analogbeamformer.

FIG. 2 is a graph of the power at the output of the summer of FIG. 1 asa function of signal.

FIG. 3 is a schematic of a digital beamformer for practicing thepreferred embodiment.

FIG. 4 is a timing diagram useful in illustrating the preferredembodiment.

FIG. 5 is a flow diagram of the method of the preferred embodiment.

FIG. 6 is a graph of the weighting of quantization errors for a uniformline array.

FIGS. 7, 8, and 9 are flow charts presenting more detail of the methodof the preferred embodiment.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 illustrates a prior art analog beamformer wherein a group ofhydrophones H₁ H₂ . . . H₄ are arrayed on a line 11. In order to sense asignal wavefront 13 at an arrival angle to the hydrophone line 11, thehydrophone output signals are delayed by appropriate amounts so thatthey add coherently, in phase. In FIG. 1, the hydrophone output signals15, 17, 19, 21 are delayed by respective analog delay elements 23, 25,27, 29. The outputs of these analog delay elements 23, 25, 27, 29 aremultiplied by respective shading coefficients a₁ . . . a₄ and then addedby a summer 31.

An illustration of the typical resultant output power of the beamformersummer 31 as a function of the incident or steering angle φ is shown inFIG. 2. The summer output is characterized by a main lobe 33, andseveral sidelobes 35. The main lobe 33 is formed about a maximumresponse axis (MRA).

FIG. 3 illustrates a digital beamformer employing a digital processor39. In this beamformer, the outputs of the respective hydrophones H₀,H₁, . . . H_(n) are sampled by respective analog to digital converters37. The successive samples are fed to a digital processor 39 forstorage, shading, and summing.

The beamformer processor 39 examines a wide variety of incident anglesφ. For each angle φ, the processor 39 selects the hydrophone outputH_(n) sampled at the time dictated by the quantized delay value for thatangle for summing.

Sampling in the digital beamformer of FIG. 3 is illustrated in furtherdetail in FIG. 4. In FIG. 4, the hydrophone outputs H_(o1), H_(o2),H_(o3), H_(o4) are sampled at intervals of T_(s). The exact samplingpoint, as dictated by the exact, unquantized delay value, is indicatedby τ_(n). Because of the periodic sampling interval of the digitalsystem, only sample H_(o1), or sample H_(o2) may be selected, ratherthan the value H_(on) dictated by the exact, unquantized delay value.Rounding of τ_(n) to either the Q₁ or Q₂ value thus affects the ultimateoutput of the beamformer.

It is the premise of the subject invention that conventional rounding ortruncation of the delays does not provide the optimum set of hydrophoneoutputs for a given steering angle. According to the preferredembodiment, a set of quantized delay values is determined by theprocessor 39 so as to minimize the derivative of the beam pattern andthereby provide a more optimum selection of the hydrophone outputs.

The derivative of the beam pattern of an arbitrary array of Nhydrophones for small quantization errors is approximately proportionalto the sum ##EQU1## where E_(n) (φ)=delay quantization error for the nthhydrophone

a_(n) =the shading coefficient for the nth hydrophone

τ(n,φ)=sum of beamformer and propagation delay for the nth hydrophonewith a target at φ

τ'(n,φ)=derivative of τ(n,φ) with respect to φ

For a given array geometry, τ(n,φ) and τ'(n,φ) are known. To minimizethe derivative of the beam pattern at a desired steering angle φ=φ_(s),the delays are chosen to minimize the weighted sum of the quantizationerrors, S(φ_(s)).

In a digital beamformer, the delay must be represented as an integermultiple of the sample interval, T_(s). In the preferred embodiment,only two quantized values of the exact delay τ(n,φ) are considered.These are the integral multiples immediately larger and immediatelysmaller than the exact value of the delay τ(n,φ). Use of only two valuesassures that the beam is nominally steered in the desired direction andthat the quantization errors are therefore small, allowing use ofequation (1) to compute S(φ).

Mathematically, the two quantized values Q₁, Q₂ of the exact delay,τ(n,φ), are represented as follows: ##EQU2## where x and x are thesmallest integer greater than x and the largest integer less than x,respectively There are therefore 2^(N) possible quantization selectionsfor use in the beamformer. The quantization scheme is to compute S(φ)from equation (1) for each of these 2^(N) combinations and utilize theset of quantized delay values for which S(φ) is smallest.

As an example, when the array is a uniformly spaced line array withspacing, d, as in FIG. 3, the delay,

τ(n,φ), is given by ##EQU3## where φ_(s) is the desired steering angleand c is the speed of sound. Then, at frequency ω, equation (1) reducesto ##EQU4## The weighting function, ##EQU5## is pre-computed and storedfor use in the evaluation of S(φ) over the 2^(N) quantization values.

The computation of the 2^(N) sums required for selection of thequantized delay set can be performed efficiently by using the fact thatthe quantization errors associated with the two quantized values of theexact delay Q1(τ_(n)) and Q2(τ_(n)), say E_(1n) and E_(2n), are relatedby

    E.sub.2n =E.sub.1n +T.sub.s                                (6)

FIG. 5 is a flow diagram illustrating the overall method of determiningthe optimum delay set. In steps 41 and 43 the actual delay values arecomputed to full arithmetic precision and then truncated to obtainQ_(1n) [τ_(n) ] and E_(1n).

In step 45, S(φ) is computed for each delay set. First, the N products,W_(n) E_(1n), are computed using the stored values of W_(n). Then theseN products are accumulated to yield the value of S(φ) for the case whenall delays are truncated, Q_(1n) (τ_(n)). All other values of S(φ) canbe computed from this initial value by appropriate addition of W_(n)T_(s), using equation (6). Consequently, if the W_(n) T_(s) areprecomputed and stored, evaluation of the 2^(N) sums requires only Nmultiplies and (N+2^(N) -1) additions. Once the smallest sum isdetermined, the associated set of delays can be calculated from thetruncated set by at most N additions of T_(s), since

    Q.sub.2 [τ.sub.n ]=Q.sub.1 [τ.sub.n ]+T.sub.s      (8)

The computation of the '2^(N) sums is implemented digitally as follows:Let A be an N-bit binary number with LSB, A_(o),

    A=[A.sub.N-1 A.sub.N-2 . . . A.sub.1 A.sub.0 ]             (8)

with A associated with the nth hydrophone. Let A_(n) =0^(n) denote theuse of Q₁ (τ_(n)) to represent τ_(n) and A_(n) =1 denote the use of Q₂(τ_(n)). Let d(A) be the decimal value of A and A(d) the binaryrepresentation of d. Then the 2^(N) sums, (4), can be represented asS_(k) (φ), k=0, 1, . . . 2^(N) -1, where the binary number A(k)indicates whether E_(1n) or E_(2n) is used in computation of the sum,i.e, if A_(n) (k)=0 use E_(1n), and if A_(n) (k)=1 used E_(2n). Clearly,the S_(o) (φ) is the value of the sum computed using the truncateddelays. A sum S_(k) (φ) can be computed from another sum, S_(m) (φ),with one addition,

    S.sub.k (φ)=S.sub.m (φ)+W.sub.n T.sub.s            (9)

if A(k) differs from A(m) by only an additional "one" in the nth bit,i.e.,

    A.sub.n (m)=0, A.sub.n (k)=1                               (10)

and

    A.sub.j (m)=A.sub.j (k), j≠n

For example, S₃ (φ) can be computed from S₂ (φ) since for N=4, A(2)=0010and A(3)=0011, implying

    S.sub.3 (φ)=S.sub.2 (φ)+W.sub.o T.sub.s

By selecting the proper ordering of calculation of the S_(k) (φ) forK=1,2, . . . N-1, all S_(k) (φ) except S_(o) (φ) can be computed fromprevious values with one addition as discussed above.

As each sum, S_(k) (φ), is computed it is compared to the smallestprevious sum (where S_(o) (φ) is used as the starting value). If S_(k)(φ) is smaller, then the associated binary word, A(k), is retained asthe current pattern, P, so that after 2^(N) sums, P is equal to someA(p) such that S_(p) (φ)>S_(k) (φ) for all p=k. In this manner,S(φ)_(min) is determined, step 47.

Finally, the delay set corresponding to S(φ)_(min) is selected for use,step 49. Using (8), the set of quantized delays to be used is then

    Q[τ.sub.n ]=Q[τ.sub.n ]+A(p)T n=0,1, . . . N-1     (12)

requiring at most N additions (because A_(n) (p) is either 0 or 1).

FIG. 6 shows the weighting function, W_(n), for typical shadingcoefficients of a line array. It can be seen that quantization errorsnear the array center or the ends of the array are not as important asthose near the half array centers in determining beam mis-steering.Errors near the array ends have little effect because they areattenuated by the shading coefficients. Those in the center have littleinfluence on steering because their effect is easily offset by thosefarther from the center. This example illustrates why conventionalquantization methods, such as rounding, may not be the best approach.For example, if delays at the array center were perfectly quantized,then pairs of hydrophones symmetrically spaced about the array centerwould have equal but opposite (in sign) quantization errors. Theseerrors, therefore, add in their effect on missteering because W_(n) isalso an odd function about the array center.

If the steering errors are small in comparison to the main lobe width,then the steering error can be approximated as ##EQU6## where dB/dφ andd² B/dφ² are the first and second derivatives of the beam pattern, B,with respect to the signal arrival angle and where φ_(s) is the desiredsteering angle. Since dB/dφ (evaluated at φ=φ_(s)) is proportional toS(φ), the reduction in steering errors resulting from the use of thisquantization method is approximately proportional to the reduction inS(φ).

The flow chart of FIGS. 7-9 illustrates a manner of implementation ofthe delay quantization scheme applicable to either softwareimplementation on a general purpose computer or in a special purposeprocessor. The number of sensors is assumed to be N, so the number ofsums, S_(k) (φ), to be calculated is 2^(N). The first loop in the delayquantization (FIG. 7) computes the exact (to the full precision of thehost computer) delay values (step 53), and then truncates them to thenumber of bits used in the beamformer (step 55). The delay calculationis exactly as performed in most sonar systems, and is not unique to thisinvention. If ##EQU7## u(φ)=unit vector in the direction the beam is tobe steered then the nth sensor delay is given by ##EQU8## where v u isthe vector inner product and c is the speed of propagation of the planewave signal. The truncation error e₁ (n), calculated in step 56, is thedifference between the truncated delay value Q₁ (τ_(n)) and the actual(full precision) value. Until N values are computed, the test 57 (n=N?)results in looping back to step 53 to compute another delay value τ_(n).

Once N delay values are calculated, the delay quantizer then computesthe initial sum, S_(o) (φ), as shown in step 59, using a table 61 ofparameters, W_(n) a_(n), which are precomputed and stored. The shadingcoefficients, a_(n), are used to produce acceptably low sidelobes in thebeam response, and the method of calculation is well-known. The arrayweighting factors are computed based upon the array geometry asdescribed above. The values for the minimum sum, S_(min), and theassociated binary pattern, A_(min), are set to S_(o) (φ) and to zero,respectively (step 63).

The second loop in the procedure (FIG. 8) computes the remaining sums inthe efficient manner described above (steps 65, 69) and compares the sumvalues to the previous minimum (step 71). When the current sum issmaller, decision path 72 is followed, the smaller sum is retained asthe minimum sum (step 73), and the associated binary pattern is savedfor later use in calculating the quantized delays. When the current sumis not smaller than the previous minimum, decision path 74 is followed.Step 73 and path 74 each lead to block 75 where the index k isincremented. A decision is made at 77 whether all source sums have beencalculated, in which case, the flow proceeds to FIG. 9.

The sequence of calculation described above is stored in an OrderingTable 67 which gives, for each sum index, k, the index, k_(s), of thesum that is to be used to calculate sum k and the binary patternassociated with sum k, A_(k). Note that this binary pattern is just thebinary representation of k, so that if the delay quantization isimplemented in binary arithmetic, the binary value of k is available andwill be exactly the pattern, A_(k). The efficiency of the calculation isbased upon the fact that the binary pattern associated with the kth sumdiffers from that of the sum used to compute it in only one binary digit(bit). Table 1 gives an example of such an Ordering Table for an arrayof 4 sensors (N=4). When the second loop is completed, the retainedbinary pattern, A_(min), is that associated with the set of delayquantizations producing the minimum sum.

                  TABLE 1                                                         ______________________________________                                        Example of Ordering Table for N = 4 (four sensors)                            K               K.sub.s                                                                             A.sub.k                                                 ______________________________________                                        1               0     0001                                                    2               0     0010                                                    3               1     0011                                                    4               0     0100                                                    5               1     0101                                                    6               2     0110                                                    7               3     0111                                                    8               0     1000                                                    9               1     1001                                                    10              2     1010                                                    11              3     1011                                                    12              4     1100                                                    13              5     1101                                                    14              6     1110                                                    15              7     1111                                                    ______________________________________                                    

The final loop (FIG. 9) uses the retained binary pattern, A_(min), tocompute the N quantized delays, Q(τ_(j)), to be used in beamforming(step 81). This uses the jth bit of the retained binary pattern,A_(min), denoted A_(min) (j). The index "j" is initially set to zero(step 79), incremented after each quantized delay calculation (step 83)and tested (step 85) to determine whether all values have been computedor whether step 81 should be repeated for the next value of "j".

A program according to the foregoing FIGS. 7-9 has been implemented onthe Digital Equipment VAX 1170. Those skilled in the art will recognizeits ready adaptability to special purpose processor circuits as known inthe art.

A technique for the quantization of delays in digital beamformers whichproduces smaller steering errors than existing quantization methods hasthus been disclosed. This is done by choosing the quantized delay valuesto minimize the derivative of the beam pattern at the desired steeringangle. Within the main lobe, the location at which this derivative iszero defines the location of the beam pattern maximum, or MaximumResponse Axis (MRA). Therefore, by minimizing the derivative, thesteering errors are reduced Although the reduction in the derivativecould theoretically be due to broadening of the main lobe, it has beenshown that if the quantization errors are small, the main lobe shape isinsensitive to these errors.

Various applications, modifications and adaptations of the justdisclosed preferred embodiment will be apparent from the foregoingdisclosure to one skilled in the art. Therefore, it is to be understoodthat, within the scope of the appended claims, the invention may bepracticed other than as specifically disclosed herein.

What is claimed is:
 1. In a digital beamformer wherein beamforming isaccomplished by summing a number of digital outputs of a plurality ofsensors, each corresponding to a quantized delay value, to produce abeam having a derivative and pattern representing beamformer output vs.steering angle, the improvement in the method of beamformingcomprising:selecting the quantized delay values to minimize thederivative of the beam pattern at a selected steering angle.
 2. Themethod of claim 1 wherein the step of selecting includes the stepsof:determining first and second quantization error values correspondingto first and second quantized delay values for each sensor; anddetermining which combination of said first and second quantizationerror values minimizes said derivative.
 3. The method of claim 2 whereinthe step of determining the first and second error quantization valuescomprises the steps of:organizing a combination of 2^(N) sets of binarynumbers, each set having a "N" bits, N being the number of hydrophones,wherein a "zero" corresponds to the first quantization error value andwherein a "one" corresponds to the second quantization error value; andordering the sets such that each set differs from the preceding one onlyby an additional "one" in the N-th bit position.
 4. The method of claim3 wherein said sensors each provide an analog output and wherein saiddigital outputs are produced by sampling said analog outputs utilizing asample interval of T_(s) and wherein the step of determining whichcombination minimizes said derivative comprises the stepsof:successively calculating the derivative of the beamformer patternS_(k) (φ) for each successive set by addition of W_(n) T_(s), whereW_(n) is the shading factor for the N-th hydrophone; and retaining theset corresponding to the smaller value of the derivative after eachsuccessive calculation.
 5. The method of claim 4 further including thesteps of:determining the set of quantized delays Q[τ_(n) ] according tothe formula Q[τ_(n) ]=Q₁ [τ_(n) ]+A_(n) (p)T_(s) where Q₁ [τ_(n) ] isthe set of binary numbers retained at the end of said step ofsuccessively calculating the derivative.
 6. A method of digitalbeamforming employing an array of hydrophones each hydrophone providingan output, the method comprising the steps of:sampling the hydrophoneoutputs at a frequency of 1/T_(s), where T_(s) is a constant sampleinterval, to produce a plurality of sampled outputs; determining firstand second quantization error values corresponding to first and secondquantized delays for each hydrophone, respectively; determining a set oferror values which minimizes the derivative of the beam pattern of saidarray of hydrophones, one member of the set corresponding to each ofsaid hydrophones and comprising either a said first quantization errorvalue or a said second quantization error value; and utilizing said setto select the sampled outputs to be used in beamforming.
 7. The methodof claim 6 wherein the step of determining first and second quantizationerror values comprises the steps of:determining the first quantizederror value by quantizing an unquantized delay value for eachhydrophone; and determining the second quantization error value byadding T_(s) times a selected factor to each first quantized delayvalue.
 8. The method of claim 6 wherein the step of utilizing said setto select the sampled outputs comprises the steps of:determining the setof quantization delays corresponding to said set of error values; andselecting the hydrophone output sampled at the sample time correspondingto each quantized delay value in said set of quantization delays.
 9. Themethod of claim 8 wherein said selected factor is the constant "one" foreach second quantized delay value determined.
 10. The method of claim 8wherein said selected factor comprises a shading factor determined foreach hydrophone.
 11. The method of claim 8 wherein said unquantizeddelay value is determined by computing the full precision actual delayvalue.
 12. The method of claim 6 wherein the step of determining saidset of error values comprises the steps of:organizing a combination of2^(N) sets of binary numbers, each set having "N" bits, N being thenumber of hydrophones, wherein a "zero" corresponds to the firstquantization error value and wherein a "one" corresponds to the secondquantization error value; ordering the sets such that each set differsfrom the preceding one only by an additional "one" in the N-th bitposition; successively calculating the derivative of the beamformerpattern S_(k) (φ) for each successive set by addition of W_(n) T_(s),where W_(n) is the shading factor for the N-th hydrophone; and retainingthe set corresponding to the smaller value of the derivative after eachsuccessive calculation.
 13. The method of claim 12 wherein the step ofutilizing includes the step ofdetermining the set of quantized delaysQ[τ_(n) ] according to the formula Q[τ_(n) ]=Q₁ [τ_(n) ]+A_(n) (p)T_(s)where Q₁ [τ_(n) ] is the set of quantized delay values corresponding tothe first error values for each hydrophone and A_(n) (p) is the set ofbinary numbers retained at the end of said step of successivelycalculating the derivative.